On a Universal Finite Type Invariant of Knotted Trivalent Graphs

Knot theory is not generally considered an algebraic subject, due to the fact that knots don’t have much algebraic structure: there are a few operations defined on them (such as connected sum and cabling), but these don’t nearly make the space of knots finitely generated. In this thesis, following an idea of Dror Bar-Natan’s, we develop an algebraic setting for knot theory by considering the larger, richer space of knotted trivalent graphs (KTGs), which includes knots and links. KTGs along with standard operations defined on them form a finitely generated algebraic structure, in which many topological knot properties are definable using simple formulas. Thus, a homomorphic invariant of KTGs provides an algebraic way to study knots. We present a construction for such an invariant. The starting point is extending the Kontsevich integral of knots to KTGs. This was first done in a series of papers by Le, Murakami, Murakami and Ohtsuki in the late 90’s using the theory of associators. We present an elementary construction building on Kontsevich’s original definition, and discuss the homomorphicity properties of the resulting invariant, which turns out to be homomorphic with respect to almost all of the KTG operations except for one, called “edge unzip”. Unfortunately, edge unzip is crucial for finite generation, and we prove that in fact no universal finite type invariant of KTGs can intertwine all the standard operations at once. To fix this, we present an alternative construction of the space of KTGs on which a homomorphic universal finite type invariant exists. This space retains ii all the good properties of the original KTGs: it is finitely generated, includes knots, and is closely related to Drinfel’d associators. The thesis is based on two articles, one published [Da] and one preprint [BD1], the second one joint with Dror Bar-Natan.Ph